Fluid Mechanics

A continuous, amorphous substance whose
molecules move freely past one another and that
has the tendency to assume the shape of its
container;
a liquid or gas
~The American Heritage® Dictionary
of the English Language,
Fourth Edition.
1.
2.
Fluid Statics – study of fluids at rest
Fluid Dynamics – study of fluids in motion
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

Mass per unit Volume
Symbols: ρ and occasionally D
Units: kg/m3, g/cm3, etc.
Object
Air
Density (kg/m3)
1.2
Osmium metal*
22,500
White-dwarf Stars
~ 1010
Neutron Stars
~ 1018
Density relative to water.
For example:

Force per unit Area
The surface area of this room is approximately 1200 ft2
(111.5 m2) and the height of the ceilings are ?. If
this room was completely filled with only air (ρ = 1.2
kg/m3) how much mass would be in the room as a
result of the air? If the air pressure in the room is 1
atm (1.013 x 105 Pa) what would be the total force
on the floor from the weight of the air?
Fluids have mass that cannot be ignored
so pressure increases with depth.
Po
ρ
This equation only applies to
situations when the density of
the fluid stays relatively constant.
It only applies to gases (which are
compressible) over short vertical
distances.
h
P
The world record for the “No-limit” free dive is 214 m
set by Herbert Nitsch in June 2007. As a result
Herbie is known as the “Deepest Man on Earth.” If
the density of ocean water is ρ = 1025 kg/m3 what
is the pressure that Herbie experienced at the
bottom of his dive? [Assume the pressure at the
surface of the water is 1 atm = 101,300 Pa]
A tank open to the atmosphere (with
atmospheric pressure p) is filled to a height L
with a liquid of density ρ as shown in the
diagram. A block of density D (D < ρ) and
dimensions x, y, and z is attached to the
bottom of the tank by a string so that its top
surface is a distance h from the surface of
the liquid.
What is the force due to pressure on the
a) top of the block?
b) bottom of the block?
c) front of the block?
A mercury barometer consists of a long glass tube,
closed at one end, that has been filled with
mercury (ρ = 13.6 x 103 kg/m3) and then inverted in
a dish of mercury. The space above the mercury
column is almost a perfect vacuum. Compute the
atmospheric pressure on a day when the height of
mercury in a barometer is 76.0 cm. [A barometer is
a device used to measure air pressure.]
h=0
Po = 0
h
mercury
P = Patm
Pressure applied to an enclosed fluid is
transmitted undiminished to every portion of
the fluid and the walls of the containing vessel.
The pressure depends only on depth; the
shape of the container does not matter.
Increasing Po in the cylinder
will increase the pressure
throughout the cylinder.
Po
ρ
h
P
Suppose a hydraulic lift has a small cylindrical piston
with radius 5 cm and a larger piston with radius 20
cm. The mass of a car placed on the larger piston
is 1000 kg.
a) If the two pistons are at the same height, what
force must be applied to the small piston to keep
the car in the air?
b) How far must the small piston move down to lift the
car through a height of 0.10 m?
If the pressure in a car tire is atmospheric pressure, the
tire is flat. The pressure has to be greater than
atmospheric pressure to support the car.
When you check the pressure and your gauge reads
32 psi (220 kPa) that means the tire pressure is 32 psi
greater than atmospheric, or really 47 lb/in2 (psi).
[Remember atmospheric pressure is ~15 psi]
A solar water-heating system uses solar panels on the roof, 12 m
above the storage tank. The pressure at the level of the
panels is 1 atm. What is the absolute pressure at the top of
the tank? [1 atm = 1.01 x 105 Pa]
What is the gauge pressure at the top of the tank?
Follow-up question: At what distance below the panels is the
gauge pressure equal to 1 atm?
When an object is completely or partially
immersed in a fluid, the fluid exerts an upward
force on the object equal to the weight of the
fluid that is displaced by the object.
This upward force is called
the buoyant force.
Fbuoy
x
mg
A tank open to the atmosphere (with
atmospheric pressure p) is filled to a height L
with a liquid of density ρ as shown in the
diagram. A block of density D (D < ρ) and
dimensions x, y, and z is attached to the
bottom of the tank by a string so that its top
surface is a distance h from the surface of
the liquid.
a) What is the total force due to pressure on
the block?
b) What is the tension in the string?
c) If the string is cut, how long would it take
for the top of the block to reach the
surface?
A hollow plastic sphere is held below the surface of a freshwater
lake ( = 1000 kg/m3) by a cord anchored to the bottom of
the bottom of the lake. The sphere has a volume of 0.650 m3
and the tension in the cord is 900 N.
a)
b)
c)
Calculate the buoyant force exerted by the water on the
sphere.
What is the mass of the sphere?
The cord breaks and the sphere rises to the surface. When the
sphere comes to rest, what fraction of its volume will be
submerged?
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Density is mass per unit volume
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Pressure is force per unit area

Pressure varies with depth

Archimedes’ Principe: fluid exerts an upward force
on the object equal to the weight of the fluid that
is displaced by the object
2 Kinds of Fluid Flow
1. Laminar Flow – every particle that passes a particular point
moves exactly along the smooth path followed by the
particles that passed that point earlier (stream line)
2. Turbulent Flow – irregular fluid flow (eddy currents)
Laminar Flow Video
1.
2.
3.
4.
The fluid is nonviscous…viscous means to have
relatively high resistance to flow
The fluid is incompressible…its density stays
constant
The fluid motion is in a steady-state meaning that
the velocity, density, and pressure at each point in
the fluid do not change in time
The fluid moves without turbulence…laminar flow
In steady state, the rate at which fluid mass moves through a
tube with a single entry point and a single exit point remains
the same, even if the cross section of the tube varies.
A1
v1
A2
v2
Δx2
Δx1
The figure below shows a portion of a pipe for oil with
rectangular cross sections. If the flow speed at the bottom is
v, what is the flow speed at the top?
When a horizontal pipe is constricted, fluid speeds up. This
means that there must be some acceleration and therefore
some net force.
anet
vwide
Fwide
Fwide = PwideA
Fnarrow
vnarrow > vwide
Fnarrow = PnarrowA
In a static fluid pressure was dependent on depth.
The same holds true for a fluid that is in motion as
long as the cross sectional area remains constant.
Lower
Pressure
h
Higher
Pressure
Combining the effects of pressure changes and gravity into the
work energy theorem results in a relationship known as
Bernoulli’s equation. In the steady-state flow of an ideal fluid
of density ρ, the following equation is true along a streamline:
When using the equation for two points on the same streamline
the equation is:
Notice the similarities to conservation of energy. It looks like
there are K and U terms in the equation.
The figure below shows a pipe with a Venturi U-tube attached to it. Fluid of
density ρ1 flows through the pipe from point X to point Y. The cross sectional
area of the pipe at point X is A1, and the cross sectional area at Y is A2. The
pressure at X is P, and the velocity of the fluid there is v. The U-tube is filled
with a fluid of density ρ2. Express your answers in terms of all of the above
values and necessary constants.
a) Determine the velocity of
the fluid at point Y.
b) Determine the pressure at
point Y.
c) Determine the pressure at
point W.
d) Find h.
The lift force that allows an airplane to take flight can be explained by Bernoulli’s
equation…or can it?
“When a solid body is placed in a
fluid flow and a nonsymmetrical
situation occurs the direction of
the force on the body does not
coincide with the direction of
the (undisturbed) flow. This
principle makes flying possible.”
High Pressure
v
F
Low Pressure
The curve ball can be explained
by Bernoulli’s equation.
FoilSimb\Ball.html
Water enters a house through a pipe with an inside diameter of
2.0 cm at a gauge pressure of 4.0 x 105 Pa (about 4 atm, or
60 lb/in2). The cold-water pipe leading to the second-floor
bathroom 5.0 m above is 1.0 cm in diameter.
a) Find the flow speed and
gauge pressure in the
bathroom when the flow
speed at the inlet pipe is 2.0
m/s.
b) How much time is required to
fill a 100 L bathtub?
A cylindrical tank of radius R is filled to a depth L with a fluid of density ρ. A hole
of radius r (less than R) is punctured in the side of the tank a distance h from
the top. Express your answers in terms of R, r, L, h, ρ, and g.
a)
What is the ratio of the rate at which the
level of the fluid in the tank is decreasing
to the speed of the fluid emerging from
the hole?
b)
What is the speed of the fluid emerging
from the hole?
c)
Determine the distance between the
base of the cylinder and the point where
the fluid strikes the floor.
d)
Another hole is punctured at a distance
h2 from the surface, where h2 is not
equal to h. Determine h2 such that the
water coming from there lands at the
same point as the fluid from the first hole.